Determining a bulk concentration of a target in a sample using a digital assay with compartments having nonuniform volumes

ABSTRACT

An embodiment of a system includes a compartment-generating device, a compartment detector, and electronic computing circuitry. The device is configured to generate compartments of a digital assay, at least one of the compartments having a respective volume that is different from a respective volume of each of at least another one of the compartments. The detector is configured to determine a number of the compartments each having a respective number of a target that is greater than a threshold number of the target. And the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to the determined number of compartments. Because such a system can be configured to estimate a bulk concentration of a target in a source from a polydisperse digital assay, the system can be portable, and lower-cost and faster, than conventional systems.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of non-provisional U.S. patentapplication Ser. No. 16/200,447, filed Nov. 26, 2018, under AttorneyDocket No. GLOB-2020058p, titled DETERMINING A BULK CONCENTRATION OF ATARGET IN A SAMPLE USING A DIGITAL ASSAY WITH COMPARTMENTS HAVINGNONUNIFORM VOLUMES, and naming first inventor HUYNH, Toan.

The entire contents of the above-referenced applications and of allpriority documents referenced in the Application Data Sheet filedherewith are hereby incorporated by reference for all purposes.

SUMMARY

Although examples of one or more embodiments, and examples of problemssolved by one or more embodiments, are described with reference tosample droplets of disparate volumes suspended in a liquid, one or moreembodiments relate generally to techniques for determining a bulkconcentration of a target in a source from a digital assay includingcompartments of disparate volumes formed in a barrier phase.

There are situations in which it is desirable to determine a bulkconcentration of a target in a source. For example, to ensure the safetyof a large number of people, a government agency may want to determine abulk concentration of a pathogen (e.g., anthrax or another infectiousagent, virus, or parasite), toxin (e.g., botulinum), or other poison(e.g., heavy metals such as lead and mercury, or chemical agents such asa nerve agent) in a municipal water supply, or may want to determine abulk concentration of an irritant (e.g., pollen or smog), in the air.The bulk concentration is typically expressed as a ratio of the numberof “pieces” (e.g., particles, molecules, cells, atoms) of the target perunit volume, or as a normalized ratio of the number of units of a volumeoccupied by the target to a reference number of units of the volume(e.g., parts per million).

FIG. 1 is a diagram that illustrates an example of an analog techniquefor determining a bulk concentration λ_(T) of a target 10 in a source12. For example, the target 10 may be the polio virus or bacteria (e.g.,E Coli or other coliform bacteria) and the source 12 may be a municipalwater supply.

Referring to FIG. 1, one or more samples 14 are taken from the source12. For example, if the source 12 is a municipal water supply, then thesample 14 may be about 10 milliliters (mL) of water from a strategicallyselected location 16 (e.g., in the middle of the body of water, or nearan input port where the water is drawn into a water-treatment facility).

Next, each sample 14 is treated with a substance, such as a reagent,that causes the sample to exhibit one or more phenomena each having arespective level related to the concentration of the target 10 in thesample. For example, a reagent added to a sample 14 can bind with themolecules of the target 10, and cause the sample to exhibit a colorhaving an intensity, saturation, hue, or shade that is related to theconcentration of the target in the sample, where the color can be causedby the bound reagent absorbing one or more wavelengths of light,luminescing one or more wavelengths of light, or absorbing one or morefirst wavelengths of light and luminescing one or more secondwavelengths of light.

Then, a technician (not shown in FIG. 1) illuminates the sample with alight source designed for activating the reagent to exhibit a color.

Next, a human technician (not shown in FIG. 1) considers the one or moreexhibited phenomena for one or more samples 14 and makes an estimation{circumflex over (λ)}_(T) of the actual bulk concentration λ_(T) of thetarget 10 in the source 12. For example, if a sample 14 exhibits alighter shade of green (left end of a shade chart 16), then thetechnician estimates the bulk concentration λ_(T) of the target 10 inthe sample 12 as “low;” conversely, if the sample exhibits a darkershade of green (right end of the shade chart), then the technicianestimates the bulk concentration λ_(T) of the target 10 in the sample 14as “high.” The technician considers the respective shade of green ofeach of one or more additional samples 14, and, based on his/herperception of the shades of green, effectively averages the shades ofgreen for all considered samples to arrive at a final estimate of thebulk concentration λ_(T). For example, the technician may characterizethe bulk concentration λ_(T) as “high,” “medium,” “low,” “dangerous,” or“safe.”

But a problem with this analog technique is that the technician cannotquantify, with any precision, his/her estimate {circumflex over (λ)}_(T)of the bulk concentration λ_(T) of the target 10 in the source 12. Thatis, with this analog technique, the technician can provide only acoarse, or “rough,” estimate {circumflex over (λ)}_(T) of the bulkconcentration λ_(T).

Unfortunately, a “rough” estimate {circumflex over (λ)}_(T) of the bulkconcentration λ_(T) is insufficient for some applications.

In another analog technique, a technician (not shown in FIG. 1) compareseach of one or more phenomena exhibited by one or more samples 14 to arespective chart, which quantifies the bulk concentration λ_(T) of thetarget 10 in the source 12 relative to a level of a respectivephenomenon, and makes an estimation {circumflex over (λ)}_(T) of thebulk concentration λ_(T) in response to the one or more charts. Forexample, the chart 16 includes four shades of green that are eachassociated with a corresponding value λ_(T_1)-λ_(T_4) of the bulkconcentration λ_(T), where the association between a shade of green andcorresponding value λ_(T_n) was previously determined using one or moretest sources having known values of the bulk concentration λ_(T) of thetarget 10. The technician compares the shade of each sample 14 with thefour shades of green in the chart 16. If, per the example shown in FIG.1, the shade of the sample 14 is between two of the shades (here thesample is between the two leftmost shades) in the chart 16, then, basedon his/her perception of how “close” the shade of the sample is to eachof the two shades, the technician interpolates the bulk concentrationλ_(T) as having an estimated value {circumflex over(λ)}_(T_Interpolated) that lies between the values λ_(T_1) and λ_(T_2)associated with the two shades in the chart (this color-shadeinterpolation is similar to the color-shade interpolation that oneperforms to determine the pH and alkalinity levels of water in aswimming pool or spa). The technician may compare, to the chart 16, therespective shade of green of each of one or more additional samples 14,and average the interpolated values of the bulk concentration obtainedfrom all compared samples to arrive at a final estimated value{circumflex over (λ)}_(T_Interpolated) of the bulk concentration λ_(T)of the target 10.

Although the latter analog technique may provide a more accurateestimate {circumflex over (λ)}_(T) of the bulk concentration λ_(T) ofthe target 10 in the source 12, this technique still depends on theabilities of a human technician to distinguish sometimes subtledifferences in the shades of a color, or in the levels of one or moreother phenomena.

FIG. 2 is a diagram that illustrates a digital technique for determininga bulk concentration λ_(T) of a target 10 in a source 12.

A digital assay 20 is formed by dividing sample into compartments (alsocalled droplets if the compartments are of a liquid) 22, each of whichis small enough (e.g., ≤−100 picoliters (pL)) such that somecompartments include the target 10, and some compartments do not includethe target. The number of compartments 22 in the digital assay 20 canrange from tens to thousands depending on the application and the amountof precision desired.

The technique is a digital technique because what is considered iswhether a compartment 22 does include at least one target 10 (an “on”compartment) or does not include at least one target (an “off”compartment). For example, as described above in conjunction with FIG.1, a reagent is added to each compartment 22, and binds to any moleculeof the target 10 in the compartment (in this example, “at least onetarget” means at least one molecule of the target or, said another way,means at least one target molecule). If the compartment 22 turns anyshade of green (i.e., the compartment includes at least one target),then the compartment is an “on” compartment; conversely, if thecompartment does not turn green (i.e., the compartment lacks anytarget), then the compartment is an “off” compartment.

Algorithms exist for generating an estimate {circumflex over (λ)}_(T) ofthe bulk concentration λ_(T) of the target 10 in the source 12 inresponse to characteristics exhibited by a digital assay, thecharacteristics including the number of “on” compartments 22, the numberof “off” compartments, and the aggregate volume of the compartments.

Because the volumes of the compartments 22 are relatively small,low-cost, portable equipment often generates the compartments havingsignificantly different volumes, where the largest compartment volume inthe digital assay 20 is, for example, approximately ten or more timesthe smallest compartment volume. A digital assay having compartmentswith such disparate volumes is called a “polydisperse digital assay.”

Unfortunately, the accuracy of existing algorithms decreasesdramatically as the uniformity of the compartment volumes decreases.Said another way, as the disparity among the compartment volumesincreases, the accuracy of the estimated bulk concentration {circumflexover (λ)}_(T) determined by existing algorithms decreases.

Consequently, for many applications, the disparity in the volumes of thecompartments 22 generated by low-cost equipment is so large thatexisting algorithms cannot yield sufficiently accurate values of{circumflex over (λ)}_(T).

Still referring to FIG. 2, to increase the accuracy of existingalgorithms, specialized equipment has been developed to generate digitalassays having compartments with more uniform volumes.

For example, a digital assay 30 has compartments 32 with approximatelyequal volumes, and such a digital assay is called a “monodispersedigital assay.”

But although a monodisperse digital assay, such as the digital assay 30,significantly increases the accuracy with which existing algorithms canestimate the bulk concentration λ_(T) of the target 10 in the source 12,the generation of a monodisperse digital assay is often beset by anumber of problems.

For example, equipment for generating a monodisperse digital assay, suchas the digital assay 30, can be expensive, bulky, and slow. Suchequipment can cost US$150,000 or more; therefore, such equipment isoften unattainable by charitable and other organizations with limitedfunds. Consequently, such organizations send out their samples to a labfor analysis, and typically wait a significant amount of time (e.g., afew weeks to a few months) for an estimate {circumflex over (λ)}_(T) ofthe bulk concentration λ_(T). Furthermore, such equipment can be on theorder of 6 feet×6 feet×2 feet; therefore, it is often unsuitable foron-site applications (e.g., on the bank of a reservoir, at a well fordrinking water). Consequently, even if an organization owns, orotherwise has access to, such equipment, transporting the sample fromthe source to the equipment increases the time for, and the cost of,obtaining an estimate {circumflex over (λ)}_(T) of the bulkconcentration λ_(T). Moreover, such equipment can take a relatively longtime, e.g., on the order of one minute, to generate each compartment ofa monodisperse digital assay. Consequently, because a monodispersedigital assay may include tens, hundreds, or even thousands ofcompartments, the throughput of such equipment is limited, and,therefore, increases the time for, and the cost of, obtaining anestimate {circumflex over (λ)}_(T) of the bulk concentration λ_(T).

Therefore, a need has arisen for a system that is smaller, lessexpensive, and faster than existing equipment, yet that is at least asaccurate as existing systems.

In an embodiment, such a system includes a compartment-generatingdevice, a compartment detector, and electronic computing circuitry. Thedevice is configured to generate compartments of a digital assay, atleast one of the compartments having a respective volume that isdifferent from a respective volume of each of at least another one ofthe compartments. The detector is configured to determine a number ofthe compartments each having a respective concentration of a target thatis greater than a threshold concentration. And the electronic circuitryis configured to determine a bulk concentration of the target in asource of the sample in response to the number.

Compared to equipment for generating a monodisperse digital assay, sucha system can be portable, lower-cost, and faster, yet can yield similaraccuracy. In an embodiment, these improvements flow from the systembeing configured to implement an algorithm that allows for accuratelyestimating the bulk concentration λ_(T) of a target in a source from apolydisperse digital assay.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram that illustrates an analog technique for determininga bulk concentration of a target in a source.

FIG. 2 is diagram that illustrates two digital techniques fordetermining a bulk concentration of a target in a source.

FIG. 3 is a diagram that illustrates a digital technique for determininga bulk concentration of a target in a source, according to anembodiment.

FIG. 4 is a diagram of a system configured to implement the digitaltechnique illustrated by FIG. 3, according to an embodiment.

FIG. 5 is a flow chart of the digital technique illustrated by FIG. 3and implemented by the system of FIG. 4, according to an embodiment.

FIG. 6 is a flow chart of the digital technique illustrated by FIG. 3and implemented by the system of FIG. 4, according to anotherembodiment.

FIG. 7 is a flow chart of a technique for characterizing thepolydisperse digital assay of FIGS. 3-4, according to an embodiment.

FIG. 8 is a flow chart of the digital technique illustrated by FIG. 3and implemented by FIG. 4, according to yet another embodiment.

FIG. 9 is a diagram of a polydisperse digital assay and a residualbarrier phase, according to an embodiment.

FIG. 10 is a diagram of a portable kit for determining a bulkconcentration of a target in a source using the digital techniqueillustrated by FIG. 3, according to an embodiment.

FIG. 11 is a diagram of the computer of FIG. 4, according to anembodiment.

DETAILED DESCRIPTION

The words “approximately,” “substantially,” “about,” and similar wordsand phrases, are used below to indicate that a quantity can be in rangeof ±10% of a value given for the quantity, and that two or morequantities can be exactly equal, or can be within ±10% of each other.Furthermore, use of such a word to describe a range b to c indicates arange of b−10%·|c−b| to c+10%·|c−b|.

FIG. 3 is a diagram that illustrates a digital technique for determininga bulk concentration λ_(T) of a target 10 in a source 12, according toan embodiment. For example, one can use the digital technique to obtainan accurate estimate {circumflex over (λ)}_(T) of the bulk concentrationλ_(T).

A container 40, such as a clear test tube of glass or plastic, holds apolydisperse digital assay 42 of compartments 44 suspended in a barrierphase 46, according to an embodiment in which a dark compartment is “on”and a light compartment is “off.” In the described example, eachcompartment 44 is a respective droplet of a liquid, such as water, andthe barrier phase 46 is another liquid, such as an oil, in which thedroplets are suspended. The combination of the droplets 44 and theliquid barrier phase 46 is an emulsion.

FIG. 4 is a diagram of the container 40, the polydisperse digital assay42 and the barrier phase 46 within the container, and a system 50configured to determine the bulk concentration λ_(T) of the target 10(FIG. 3) in the source 12 (FIG. 3) in response to at least some of thedroplets 44 of the polydisperse digital assay, according to anembodiment.

The system 50 includes a droplet analyzer 52 and a computing device 54.Both the droplet analyzer 52 and computing device 54 include electroniccircuitry that is hardwired, or is configured by software or firmware,to perform the respective functions and operates described below.

The droplet analyzer 52 includes an “on”-droplet detector and counter56, a droplet counter 58, and a droplet-volume determiner 60. The“on”-droplet detector and counter 56 includes electronic circuitry andone more optical sensors configured to detect, and to determine thenumber of, “on” droplets 44 in the polydisperse digital assay 42. Thedroplet counter 58 includes electronic circuitry and one or more opticalsensors (one or more of which may be shared with the on-droplet detectorand counter 56) configured to determine the total number of “on” and“off” droplets 44 in the polydisperse digital assay 42. And thedroplet-volume detector 60 includes electronic circuitry and one or moreoptical sensors (one or more of which may be shared with the“on”-droplet detector and counter 56 or the droplet counter 58)configured to measure, or otherwise to determine, the respective volumeof each of the droplets 44.

The computing device 54 can be any suitable computer, such a laptop, atablet, or a smart phone, that includes one or more microprocessors ormicrocontrollers.

FIG. 5 is a flow chart 70 of an algorithm for determining a bulkconcentration λ_(T) of a target 10 in a source 12, according to anembodiment.

Referring to FIGS. 3-5, the algorithm represented by the flow chart 70,and operation of the system 50 while implementing the algorithm, aredescribed, according to an embodiment in which both the compartments 44and the barrier phase 46 are liquids such that the compartments 44 aredroplets suspended in the liquid barrier phase.

First, at a step 72, a technician (not shown in FIGS. 3-5) generates thedroplets 44 of different volumes (e.g., in a range of approximately 1pL-100s pL) from a sample of the source 12 to form the polydispersedigital assay 42. For example, the technician adds an indicated volumeof the barrier-phase liquid 46, such as an oil, to the container 40,adds an indicated sample volume of the source 10 to the barrier-phaseliquid in the container, plugs the top of the container, and shakes thecontainer to form an emulsion of the droplets 44 suspended in thebarrier-phase liquid. Further in example, the container 40 includes ameasurement line (not shown in FIGS. 3-5) to indicate the volume of thebarrier-phase liquid to be added, and the technician uses a measurementdevice, such as an “eye” dropper, to obtain and measure the volume ofthe sample. Alternatively, the container 40 may include anothermeasurement line (not shown in FIGS. 3-5), higher up on the containerthan the barrier-phase measurement line, to indicate the volume of thesample to be added after the barrier-phase liquid is added. Stillfurther in example, the technician adds to the container 40, before thetechnician shakes it, a reagent for rendering target-carrying droplets44 luminescent as described above in conjunction with FIGS. 1-2. Usingthis “shake-and-bake” technique, the technician can generate thepolydisperse digital assay 42 in a matter of a few ones to tens ofseconds, and in no more than a few minutes even if the time for settingup the system 50 and obtaining the sample is included.

Next, at a step 74, the “on”-droplet detector and counter 56 determinesthe number a of “on” droplets 44 in the polydisperse digital assay 42.For example, the counter 56 can include a combination illuminationdevice and image-capture device, such as a light source and a smallcamera, which the technician holds up near, or against, the container40. The illumination device illuminates the droplets 44 so that dropletsincluding the target luminesce a color having shades respectivelycorresponding to the concentrations of the target in the droplets. Thetechnician then presses a button on the device, or a virtual buttondisplayed by the computer 54, to capture an image of the droplets 44 inthe container while the droplets including the target are luminescing.Using conventional image-processing techniques, the computer 54 analyzesthe image, detects the droplets 44, and determines whether each detecteddroplet 44 is “on” or “off” by determining, for each droplet, whetherthe number of targets (or, said another way, the number of the target)within the droplet exceeds a threshold number (e.g., one targetmolecule, five target molecules, ten target molecules). For example, anoptical signal that the target luminesces has a property (e.g.,intensity, color, color shade) indicative of the number of targetswithin a droplet 44, and the computer 54 determines whether the numberof targets within the droplet exceeds the threshold number bydetermining whether the property of the target-related optical signalexceeds (or is below) a signal-property threshold. Further in example,the computer 54 compares a shade of the color (e.g., green), or anopacity, of a droplet 44 to a threshold shade or opacity, determinesthat the droplet is “on” if the level of the shade or opacity is greaterthan or equal to the threshold, and determine that the droplet is “off”if the level of the shade or opacity is less than the threshold.Alternatively, the technician uses the “on”-droplet detector and counter56 to capture multiple images of the droplets 44 from differentorientations relative to the container 40 so that the computer 54 isable to detect droplets that might otherwise be obscured by otherdroplets in a single image.

Then, at a step 76, the computer 54 generates an estimate {circumflexover (λ)}_(T) of the bulk concentration λ_(T) of the target 10 in thesource 12 in response to the number a of “on” droplets 44 in thecontainer 40. For example, as described below in conjunction with FIG.6, the computer 54 executes one or more equations to solve for{circumflex over (λ)}_(T) in response to a.

The system 50, and the algorithm that the system implements, provide oneor more advantages over existing systems and techniques for determininga bulk concentration of a target in a source. For example, the container40 and the barrier phase 46 are configured to provide inexpensive,on-site, and fast generation of the digital assay 42. Furthermore, thesystem 50 is configured to provide inexpensive, on-site, and fastestimation of the bulk concentration λ_(T) of the target 10 in thesource 12 even in response to a polydisperse digital assay 42 havingcompartments 44 of disparate volumes.

Still referring to FIGS. 3-5, alternate embodiments of the system 50 andof the above-described algorithm are contemplated. For example, one ormore steps can be added to the algorithm, and one or more of theabove-described steps can be omitted from the algorithm. Furthermore,one or more components can be added to the system 50, and one or more ofthe above-described components can be omitted from the system. Moreover,use of a reagent may be omitted if, for example, the target luminesceswithout the reagent, or if the system 50 can determine, in a manner thatdoes not involve use of a reagent, whether the number of targets in acompartment 44 exceeds the threshold number by determining whether aproperty of a target-related signal (e.g., color, color shade) exceeds asignal-property threshold. In addition, the computer 54 may perform oneor more functions and operations attributed to the droplet analyzer 52,and the droplet analyzer may perform one or more functions andoperations attributed to the computer.

Furthermore, a cloud server may complement, or replace, the computer 54.Moreover, the system 50 and the above-described algorithm yield similarresults and advantages for a monodisperse digital assay. In addition,embodiments described below in conjunction with FIGS. 6-10 may beapplicable to the system 50 and to the above-described algorithm.

FIG. 6 is a flow chart 80 of a digital-variable-volume (DVV) algorithmfor determining a bulk concentration λ_(T) of a target 10 (FIG. 3) in asource 12 (FIG. 3), according to an embodiment. The DVV algorithm issuitable for situations in which the system 50 includes thedroplet-volume determiner 60, or in which the respective volumes of the“on” droplets 44, and the aggregate volume of all the droplets,otherwise can be determined.

Referring to FIGS. 3-4 and 6, the DVV algorithm represented by the flowchart 80, and operation of the system 50 while implementing thealgorithm, are described according to an embodiment in which both thecompartments 44 and the barrier phase 46 are liquids such that thecompartments 44 are droplets suspended in the barrier phase to form anemulsion.

First, at a step 82, a technician (not shown in FIGS. 3-4 and 6)generates the droplets 44 of disparate volumes (e.g., in a range ofapproximately 1 pL-100s pL) from a sample of the source 12 to form thepolydisperse digital assay 42. For example, the technician may form thepolydisperse digital assay 42 using a method that is the same as, orthat is similar to, the “shake-and-bake” method described above inconjunction with step 72 of FIG. 5.

Next, at a step 84, the “on”-droplet detector and counter 56 determinesthe number a of “on” droplets 44 in the polydisperse digital assay 42.For example, the “on”-droplet detector and counter 56 may determine thenumber a using a method that is the same as, or that is similar to, themethod described above in conjunction with step 74 of FIG. 5.

Then, at a step 86, the droplet-volume determiner 60 determines therespective volume v_(i) of each of the detected “on” droplets 44, and,if necessary, determines the aggregate volume V_(Total) of the droplets44 by summing the respective volumes of the detected “on” and “off”droplets. For example, to determine the respective volume v_(i) of eachof the a “on” droplets 44, the determiner 60 analyzes the one or moreimages that the system 50 captured at the step 84 using a conventionaldroplet-volume-determining algorithm. To determine the aggregate volumeV_(Total), the determiner 60 also determines the volumes of the “off”droplets 44 in the same way that the determiner determines the volumesv_(i) of the “on” droplets, sums the volumes of the “off” droplets withthe volumes v_(i) of the “on” droplets, and sets V_(Total) equal to thedetermined sum. Alternatively, the droplet-volume determiner 60 isconfigured to operate as described above but is part of, or is otherwiseincluded in, the computer 54 instead of the droplet analyzer 52. In yetanother alternative, because the aggregate volume of the sample is thesame as the aggregate volume V_(Total) of the droplets 44, and becausethe volume of the sample is known per step 82, the technician entersinto the computer 54 the volume of the sample, and the computer setsV_(Total) equal to the entered sample volume.

Next, at a step 88, the computer 54 solves for the estimated bulkconcentration {circumflex over (λ)}_(T) of the target 10 in the source12 according to the following equation:

$\begin{matrix}{{\underset{i = 1}{\sum\limits^{a}}\frac{v_{i}}{1 - e^{{- v_{i}}{\overset{\hat{}}{\lambda}}_{T}}}} = V_{Total}} & (1)\end{matrix}$

A derivation and explanation of equation (1) is included below.

In an ideal example, each “on” droplet 44 would contain one and only oneof the target such that the computer 54 could determine the estimatedbulk concentration {circumflex over (λ)}_(T) from the number a of “on”droplets 44 divided by the volume V_(Total) of the sample 14, where awould also equal the number of targets in the sample.

But because in an actual, non-ideal, example each “on” droplet 44 maycontain more than one of the target 10, determining the estimated bulkconcentration {circumflex over (λ)}_(T) from a/V_(Total) may lead to anerror caused by an undercounting of the number of the target in thesample 14.

To reduce or eliminate such an undercounting error where the respectivenumber of the target in one or more of the “on” droplets 44 of thesample 14 is unknown, the computer 54 is configured to use equation (1)to estimate the bulk concentration λ_(T) of the target in the source 12.

For each “on” droplet 44, equation (1) includes a respective expressionfor the probability that the droplet includes at least one of thetarget, the probability being dependent on, and, therefore, therespective expression including, the respective volume of the droplet.

Consequently, equation (1) not only effectively accounts for thepossibility that each of one or more of the “on” droplets 44 containsmore than one of the target 10, equation (1) also effectively accountsfor a larger “on” droplet 44 being more likely than a smaller “on”droplet to contain more than one of the target.

The system 50, and the DVV algorithm that the system 50 is configured toimplement, provide one or more advantages over existing systems andtechniques. For example, the container 40 and binary phase 46 providefor inexpensive, on-site, and fast generation of the digital assay 42.Furthermore, the system 50 provides for inexpensive, on-site, fast, andaccurate estimation of a bulk concentration λ_(T) of a target 10 in asource 12 even in response to a polydisperse digital assay 42 havingdroplets 44 of disparate volumes.

Still referring to FIGS. 3-4 and 6, alternate embodiments of the system50 and the DVV algorithm are contemplated. For example, instead of the“on”-droplet detector and counter 56 determining the number a of “on”droplets 44, the technician may count the number a of “on” droplets andenter the number a into the computer 54. Moreover, instead of thedroplet-volume determiner 60 determining the volumes v_(i) of the “on”droplets 44, the technician may use a device, such as a ruler ormicroscope, to estimate the volumes v_(i), and then enter these volumesinto the computer 54. Moreover, the system 50 and the above-describedalgorithm yield similar results and advantages for a monodispersedigital assay. In addition, alternate embodiments described above inconjunction with FIGS. 3-5 and below in conjunction with FIGS. 7-11 maybe applicable to the system 50 and the DVV algorithm.

FIG. 7 is a flow chart 100 of an algorithm for characterizing thevolumes of the compartments (e.g., droplets) 44 generated with thecontainer 40 and barrier phase 46 of FIG. 4, according to an embodiment.

FIG. 8 is a flow chart 110 of a digital-variable-volume-approximation(DVVA) algorithm for determining a bulk concentration λ_(T) of a target10 (FIG. 3) in a source 12 (FIG. 3), according to an embodiment. TheDVVA algorithm is suitable for situations in which the system 50 lacksthe droplet-volume determiner 60, or in which the respective volumes ofthe “on” droplets 44, and the aggregate volume of all the droplets, areotherwise unknown.

Referring to FIGS. 3-4 and 7, the compartment-volume characterizingalgorithm is described, according to an embodiment in which thecompartments 44 are droplets.

At a step 102, a technician (not shown in FIG. 3-4 or 7) generates a setof test droplets of different volumes (e.g., in a range of approximately1 pL-100 pL) from a test sample that is similar to a sample of thesource 12 of FIG. 3 to form a test polydisperse digital assay; forexample, if the intended source 12 is a body of water, then thetechnician may use a sample of water. The technician adds an indicatedvolume of the barrier-phase liquid 46, such as an oil, to the container40, adds an indicated volume of the test sample to the barrier-phaseliquid in the container, plugs the top of the container, and shakes thecontainer to form an emulsion of test droplets suspended in thebarrier-phase liquid. The container 40 may include a measurement line(not shown in FIG. 3-4 or 7) to indicate the volume of the barrier-phaseliquid to be added, and the technician may use a measurement device,such as an “eye” dropper, to add an indicated volume of the test sample.Alternatively, the container may include another measurement line (notshown in FIG. 3-4 or 7), higher up on the container than thebarrier-phase measurement line, to indicate the volume of the testsample to be added after the barrier-phase liquid is added.

Next, at a step 104, the droplet counter 58, or a similar counter,determines the number m of test droplets, and at a step 106, thedroplet-volume determiner 60, or a similar determiner, determines arespective volume v_(i_characterized) for each of the m test droplets.

Then, at a step 108, the computer 54 stores the number m of testdroplets, and stores the volumes v_(i_characterized) of the testdroplets, in a memory (not shown in FIG. 3). Alternatively, the number mand the corresponding volumes v_(i_characterized) can be stored inanother memory from which the computer 54 is configured to download thevalues of m and v_(i_characterized).

The theory behind generating the characterized number m and thecharacterized volumes v_(i_characterized) is that similar containers,sample substances, and barrier phases will generate similar values for mand v_(i_characterized) such that the values of m andv_(i_characterized) can be used to determine a bulk concentration λ_(T)of the target 10 (FIG. 3) in the sample 12 (FIG. 3) in situations wherethe volumes v_(i) of the actual droplets 44 cannot be determined or areotherwise unknown. That is, the statistical dependence between theactual droplet volumes v_(i) and the characterized droplet volumesv_(i_characterized) is high enough that, as described below, the numberm and volumes v_(i_characterized) of the test droplets can be used todetermine the actual bulk concentration λ_(T) of the target 10 in thesource 12 in a situation where the volumes v_(i) of the actual droplets44 are unknown.

Still referring to FIG. 7, alternative embodiments of thedroplet-characterization algorithm are contemplated. For example, thesteps 102-106 can be repeated any suitable number of times to generatesets of test values for m and v_(i_characterized), and the computer 54,or a similar computer, can calculate the final values of m andv_(i_characterized) by interpolating values from one or more of the setsof test values.

Referring to FIGS. 3-4 and 8, the DVVA algorithm represented by the flowchart 110, and operation of the system 50 while implementing the DVVAalgorithm, are described according to an embodiment in which both thecompartments 44 and the barrier phase 46 are liquids such that thecompartments 44 are droplets suspended in the barrier phase to form anemulsion.

First, at a step 112, a technician (not shown in FIGS. 3-4 and 8)generates the droplets 44 of disparate volumes (e.g., in a range ofapproximately 1 pL-100 pL) from a sample of the source 12 to form thepolydisperse digital assay 42. For example, the technician may form thepolydisperse digital assay 42 using a method that is the same as, orsimilar to, the “shake-and-bake” method described above in conjunctionwith step 72 of FIG. 5.

Next, at a step 114, the “on”-droplet detector and counter 56 determinesthe number a of “on” droplets 44 in the polydisperse digital assay 42.For example, the on-droplet detector and counter 56 may determine thenumber a using a method that is the same as, or similar to, the methoddescribed above in conjunction with step 74 of FIG. 5.

Then, at a step 116, the droplet counter 58 determines the number n ofall droplets 44 (i.e., the sum of the “on” and “off” droplets) in thepolydisperse digital assay 42 in the container 40. For example, todetermine the number n of all droplets 44, the droplet counter 58analyzes the one or more images that the system 50 captured at the step114 using a conventional droplet-counting algorithm. Alternatively, thedroplet counter 58 counts only the number b of “off” droplets 44, andadds this number to the number a of “on” droplets, to generate n=a+b.Alternatively, the droplet counter 58 operates in a similar manner butis part of, or is included in, the computer 54 instead of being part of,or included in, the droplet analyzer 52.

Next, at a step 118, the computer 54 solves for the estimated bulkconcentration {circumflex over (λ)}_(T) of the target 10 in the source12 according to the following equation:

$\begin{matrix}{{{\frac{1}{m}\left( {\underset{i = 1}{\sum\limits^{m}}e^{{- \nu_{i_{-}{characterized}}}{\overset{\hat{}}{\lambda}}_{T}}} \right)} - 1 + \frac{a}{n}} = 0} & (2)\end{matrix}$

where m is the characterized number of droplets and eachv_(i_characterized) is the characterized volume of a respective one ofthe m droplets as described above in conjunction with FIG. 7.

A derivation and explanation of equation (2) is included below.

The system 50, and the DVVA algorithm that the system is configured toimplement, provide one or more advantages over existing systems andtechniques. For example, the container 40 and barrier phase 46 providefor inexpensive, on-site, and fast generation of the digital assay 42.Furthermore, the system 50 provides for inexpensive, on-site, and fastestimation of a bulk concentration λ_(T) of the target 10 in the source12 even in response to a polydisperse digital assay 42 having droplets44 of disparate volumes that are unknown.

Still referring to FIGS. 3-4 and 8, alternate embodiments of the system50 and the DVVA algorithm are contemplated. For example, from the numberm and volumes v_(i_characterized) of the test droplets described abovein conjunction with FIG. 7, one may determine the probability densityfunction ƒ(v) of the volumes of the test droplets, and solve for theestimated bulk concentration {circumflex over (λ)}_(T) of the target 10in the source 12 according to the following equation:

$\begin{matrix}{\frac{a}{n} = {1 - {\int_{- \infty}^{\infty}{e^{{- {\overset{\hat{}}{\lambda}}_{T}}\nu}{f(v)}dv}}}} & (3)\end{matrix}$

Moreover, alternate embodiments described above in conjunction withFIGS. 3-7 and below in conjunction with FIGS. 9-11 may be applicable tothe system 50 and the DVVA algorithm.

FIG. 9 is a diagram of the polydisperse digital assay 42 in thecontainer 40, according another embodiment.

If the barrier phase 46, e.g., an oil, has a higher density than thedroplets 44, then the droplets may “float” over a residual region 120 ofthe barrier phase that is devoid of droplets as shown in FIG. 9.

Conversely, if the barrier phase 46, e.g., an oil, has a lower densitythan the droplets 44, then the droplets may “sink” to the bottom of thecontainer 40 such that the residual region 120 of the barrier phase thatis devoid of droplets “floats” over the droplets (the residual region“floating” over the droplets is not shown in FIG. 9).

FIG. 10 is a diagram of a digital-assay-generator-and-analyzer kit 130,according to an embodiment. The combination of the kit 130, a portablecomputer (e.g., laptop, tablet, smart phone), and one or more of thealgorithms described above in conjunction with FIGS. 5-8 provides forinexpensive, on-site, and fast generation of the digital assay 42, andprovides for inexpensive, on-site, and fast estimation of a bulkconcentration λ_(T) of a target 10 (FIG. 3) in a source 12 (FIG. 3) inspite of the polydisperse digital assay having droplets 44 of disparatevolumes that may be unknown. For example, the kit 130 may costapproximately $30-$100, the amount of the barrier phase 46 required togenerate each digital assay 42 may cost approximately $1 or less, thecombined weight of the kit 130 and the computer 54 may be approximately3 pounds (lbs.) to 10 lbs., and the total test time (from collection ofa sample 14 to the computer 54 rendering an estimated bulk concentration{circumflex over (λ)}_(T)) may be approximately 20 minutes to 80minutes.

In addition to the container 40 and the droplet analyzer 52, the kit 130includes a container stopper 132, a re-openable and re-closable package(e.g., a screw-top bottle) 134 of the barrier phase 46, a re-openableand re-closable optional package (e.g., a screw-top bottle) 136 of areagent, a dropper 138, and a non-transitory computer-readable medium140. The stopper 132 is configured to form a liquid-tight seal at theopening of the container 40 to allow shaking of the container to formthe polydisperse digital assay 42. The dropper 138 allows a technicianto transfer the barrier phase 46 and the reagent from their respectivepackages 134 and 136 to the container 40, and allows a technician toobtain a liquid sample (e.g., water) from a source (e.g., reservoir) andto transfer the sample to the container. And the computer-readablemedium is a suitable non-volatile memory that stores programinstructions that, when executed by a portable computer, cause thecomputer to implement one of the algorithms described above inconjunction with FIGS. 5-8.

Still referring to FIG. 10, alternate embodiments of the kit 130 arecontemplated. For example, the kit 130 may include a carrying case inwhich all of the other system components may be stored and carried.Furthermore, embodiments describe above in conjunction with FIGS. 1-9and below in conjunction with FIG. 11 may be applicable to the kit 130.

FIG. 11 is a block diagram of the computer 54 of FIG. 4, according to anembodiment.

The computer 54 includes computing circuitry 150, one or more inputdevices 152, one or more output devices 154, and one or moredata-storage devices 156.

The computing circuitry 150 includes circuitry that is configured toperform various functions and operations, such as the functions andoperations described above in conjunction with FIGS. 3-8 and 10. Forexample, the computing circuitry 150 includes a microprocessor ormicrocontroller that is hardwired or configured with firmware, or thatexecutes software, to perform the above-described functions andoperations.

The one or more input devices 152 are configured to allow an operator ordevice to provide data or other information or signals to the computer54. Examples of an input device 152 include a keyboard, mouse, touchscreen, audible or voice-recognition component, the droplet analyzer 52(FIG. 4), and so on

The one or more output devices 154 are configured to provide data fromthe computing circuitry 150 to an operator or device in a suitable form,or to perform a function or operation under control of the computingcircuitry 150. Examples of an output device 154 include a printer, videodisplay, audio output components, the droplet analyzer 52 (FIG. 4), andso on.

The one or more data-storage devices 156 are configured to store data onor to retrieve data from volatile or non-volatile storage media (notshown). Examples of a data-storage device 156 include a magnetic disk, aFLASH memory, other types of solid state memory such as a random-accessmemory (RAM, SRAM, DRAM, USB “stick”), a ferro-electric memory, a tapedrive, an optical disk like a compact disk and a digital versatile disk(DVDs), and so on.

Still referring to FIG. 11, alternate embodiments of the computer 54 arecontemplated. For example, the computer 54 may omit one or more of theabove-described devices, and may include one or more other devices.

Derivation and Explanation of Equations (1) and (2) General Definitionsand Assumptions

In digital assays, the targets (molecules, cells, etc.) in the bulksample are randomly distributed into many compartments. A compartmentwith one or more targets gives a signal (e.g., fluorescent intensityafter nucleic acid amplification), and is called an “on” compartment. Acompartment without targets does not provide a signal, and is called an“off” compartment. Targets are distributed into compartments followingthe Poisson distribution. An assay system, such as the system 50 of FIG.4, can detect and count the number of “on” compartments (but not thenumber of targets per compartment).

For each assay, the bulk concentration needs to be calculated using acertain inference method. The digital-variable-volume (DVV) anddigital-variable-volume-approximation (DVVA) methods are based onmaximum likelihood estimation; the concentration estimate is the onethat maximizes the likelihood of observing a certain experimentalresult. The choice of maximum likelihood estimation was inspired by itsuse in multivolume digital PCR (where each assay utilizes a handful ofpredetermined, precisely controlled volumes), which has been inspired bylimiting dilution assays for microorganism counting. In particular, animportant feature is that results from different volumes are readilycombined by way of multiplying the likelihoods. Below, are derived theexpressions used to calculate the concentration estimates and thestandard errors using the maximum likelihood framework. The termsrelevant to the descriptions of the DVV and DVVA methods are describedin Table 1.

TABLE 1 Definitions of mathematical symbols. Symbol Definition{circumflex over (x)} Estimator of a particular parameter denoted as x

x 

Expectation of the quantity x over a certain distribution λ_(T) Bulkconcentration (number of targets/unit volume)

_(T) Estimator of λ_(T) (inferred from the assay result) Λ ≡ ln(λ_(T))Natural log of bulk concentration

Estimator of Λ (inferred from the assay result)

Standard error of 

Λ₀ A with smallest 

N Total number of compartments V_(total) Total volume of compartments ANumber of ON compartments A ≡ {v₁, v₂, . . . v_(a)} Set of volumes of ONcompartments b ≡ n - a Number of OFF compartments$V_{b\overset{def}{=}}{\sum\limits_{i = 1}^{b}v_{i}}$ Total Volume ofOFF compartments M Number of pre-measured (test or characterization)volumes M ≡ {v₁, v₂, . . . v_(m)} Set of pre-measured (test orcharacterization) volumes f(v) Volume probability density function μ_(v)Mean volume σ_(v) Standard deviation of volume μ_(ln)V Geometric mean ofvolume W Product logarithm function (also known as Lambert W function)

Begin by calculating the probability that a particular compartment turns“on” given the volume and bulk concentration (equation (a)). It is thesame as the probability of having more than one target in thecompartment, based on the Poisson distribution with the mean of vλ_(T).This probability is useful in subsequent derivation steps.

$\begin{matrix}{{p_{each}\left( {\lambda_{T},v} \right)} = {{1 - {{Prob}\mspace{11mu}({notargets})}} = {{{1 - \frac{\left( {v\lambda_{T}} \right)^{k}e^{{- v}\lambda_{T}}}{k!}}❘_{k = 0}} = {1 - e^{{- v}\lambda_{T}}}}}} & (a)\end{matrix}$

Digital Variable Volume (DVV)

The likelihood l(λ_(T)) of observing a certain assay result, i.e.,particular numbers of “on” and “off” compartments (a and b,respectively) with the associated volumes is the product of individuallikelihoods calculated using equation (a).

Π_(i=1) ^(a) p _(each)(λ_(T) ,v ₁)Π_(i=1) ^(b)[1−p _(each)(λ_(T) ,v_(i))]=Π_(i=1) ^(a)(1−e ^(−v) ^(i) ^(λ) ^(T) )Π₁ ^(b) e ^(−v) ^(i) ^(λ)^(T)   (b)

The value of λ_(T) that maximizes l(λ_(T)) is then found. Use thenatural logarithm of the concentration (Λ≡ln(λ_(T))) and theloglikelihood function (L(Λ)≡ln(λ_(T))) to conveniently calculate thestandard errors and enforce the requirement for positive concentrations.The calculation of the standard error is also more appropriate for Λthan for λ_(T) because the distribution of Λ is less skewed. Therefore,the goal is now finding the Λ value that maximizes L(Λ). The expressionfor L(Λ) and the first and second derivatives are shown below.

$\begin{matrix}\begin{matrix}{{L(\Lambda)} = {{\sum\limits_{i = 1}^{a}{\ln\;\left( {1 - e^{{- v_{i}}e^{\Lambda}}} \right)}} - {e^{\Lambda}{\sum\limits_{i = 1}^{b}v_{i}}}}} \\{= {{\sum\limits_{i = 1}^{a}{\ln\;\left( {1 - e^{{- v_{i}}e^{\Lambda}}} \right)}} - {e^{\Lambda}\left( {V_{total} - {\sum\limits_{i = 1}^{a}v_{i}}} \right)}}} \\{= {{\sum\limits_{i = 1}^{a}\left\lbrack {{\ln\;\left( {1 - e^{{- v_{i}}e^{\Lambda}}} \right)} + {v_{i}e^{\Lambda}}} \right\rbrack} - {V_{total}e^{\Lambda}}}}\end{matrix} & (c) \\{{L^{\prime}(\Lambda)} = {e^{\Lambda}\left( {{\sum\limits_{i = 1}^{a}\;\frac{v_{i}}{1 - e^{{- v_{i}}e^{\Lambda}}}} - V_{total}} \right)}} & (d) \\{{L^{''}(\Lambda)} = {{e^{\Lambda}\left( {{\sum\limits_{i = 1}^{a}\;\frac{v_{i}}{1 - e^{{- v_{i}}e^{\Lambda}}}} - V_{total}} \right)} - {e^{2\Lambda}{\sum\limits_{i = 1}^{a}\frac{v_{i}^{2}{e^{- v_{i}}}^{e^{\Lambda}}}{\left( {1 - e^{{- v_{i}}e^{\Lambda}}} \right)^{2}}}}}} & (e)\end{matrix}$

To calculate {circumflex over (Λ)}, the root of the first derivative(equation (d)) is determined, i.e., equation (1), which is repeatedbelow, is solved.

$\begin{matrix}{{\underset{i = 1}{\sum\limits^{a}}\frac{v_{i}}{1 - e^{{- v_{i}}{\overset{\hat{}}{\lambda}}_{T}}}} = V_{Total}} & (1)\end{matrix}$

Plugging L′(Λ)=0 into equation (e) gives L″(Λ)<0. So the Λ value foundusing equation (1) indeed maximizes L(Λ). Also, using the derivatives atΛ, the standard error of Λ also can be calculated using the observedFisher information L″(Λ).

$\begin{matrix}{\sigma_{\overset{\hat{}}{\Lambda}} = {\sqrt{variance} = {\sqrt{\frac{1}{- {L^{''}\left( \hat{\Lambda} \right)}}} = {1\text{/}\sqrt{e^{2\overset{\hat{}}{\Lambda}} \cdot {\underset{i = 1}{\sum\limits^{a}}\frac{v_{i}^{2}{e^{- v_{i}}}^{\hat{\Lambda}}}{\left( {1 - {e^{- v_{i}}}^{\hat{\Lambda}}} \right)^{2}}}}}}}} & (f)\end{matrix}$

This σ_({circumflex over (Λ)}) can be used to calculate the confidenceinterval. Calculating σ_({circumflex over (Λ)}) using the expectedFisher information is not feasible because the volume distribution isunknown. In fact, to implement the DVV technique, the volumedistribution is not required and need not be the same from oneexperiment to another.

Digital Variable Volume Approximation (DVVA)

In general, the probability a compartment turns ON can be calculatedusing the volume distribution (specified by the probability densityfunction ƒ(v)).

p _(on)(λ^(T))=∫ƒ(v)p _(each)(λ_(T) ,v)dv=∫ƒ(v)(1−e ^(−vλ) ^(T))dv=1−∫ƒ(v)e ^(−vλ) ^(T) dv  (g)

Previously, ƒ(v) has been chosen to follow the gamma distribution ortruncated normal distribution. However, in practice, ƒ(v) may not bedescribed by a simple function. And even when that is true, a set ofpre-measured volumes (M as in Table 1) still needs to be experimentallyobtained to characterize ƒ(v). Therefore, for the DVVA technique, a setof separately measured volumes, M, is used instead of ƒ(v).

$\begin{matrix}{{p_{on}(\lambda)} = {{1 - {\underset{i = 1}{\sum\limits^{m}}{\frac{1}{m}\left( e^{{- v_{i}}\lambda_{T}} \right)}}} = {1 - {\frac{1}{m}\left( {\underset{i = 1}{\sum\limits^{m}}e^{{- v_{i}}\lambda_{T}}} \right)}}}} & (h)\end{matrix}$

The likelihood function can then be obtained using the binomialdistribution (for the case of a ON compartments out of n compartmentswith the probability of p_(on)(λ).

$\begin{matrix}{{l\left( \lambda_{T} \right)} = {{\begin{pmatrix}n \\a\end{pmatrix}{p_{ON}^{a}\left( {1 - p_{ON}} \right)}^{n - a}} = {{\begin{pmatrix}n \\a\end{pmatrix}\left\lbrack {1 - {\frac{1}{m}\left( {\underset{i = 1}{\sum\limits^{m}}e^{{- v_{i}}\lambda_{T}}} \right)}} \right\rbrack}^{a}\left\lbrack {\frac{1}{m}\left( {\underset{i = 1}{\sum\limits^{m}}e^{{- v_{i}}\lambda_{T}}} \right)} \right\rbrack}^{n - a}}} & (i)\end{matrix}$

As motivated above, the loglikelihood function can be calculated withthe change of variable Λ≡ln(λ_(T)), and subsequently, its first andsecond derivatives.

$\begin{matrix}{{L(\Lambda)} = {{(a){\ln\left( {1 - \frac{\underset{i = 1}{\sum\limits^{m}}{e^{- v_{i}}}^{e^{\Lambda}}}{m}} \right)}} + {\left( {n - a} \right){\ln\left( \frac{\underset{i = 1}{\sum\limits^{m}}{e^{- v_{i}}}^{e^{\Lambda}}}{m} \right)}} + {\ln\;\begin{pmatrix}n \\a\end{pmatrix}}}} & (j) \\{{L^{\prime}(\Lambda)} = {\frac{\begin{matrix}\left( {\frac{a}{n} - 1 + {\frac{1}{m}{\underset{i = 1}{\sum\limits^{m}}{e^{- v_{i}}}^{e^{\Lambda}}}}} \right) \\{\frac{n}{m}{\underset{i = 1}{\sum\limits^{m}}{v_{i}{e^{\Lambda - v_{i}}}^{e^{\Lambda}}}}}\end{matrix}}{\left( {1 - {\frac{1}{m}{\underset{i = 1}{\sum\limits^{m}}{e^{- v_{i}}}^{e^{\Lambda}}}}} \right)\frac{1}{m}{\underset{i = 1}{\sum\limits^{m}}{e^{- v_{i}}}^{e^{\Lambda}}}} = \frac{\begin{bmatrix}{\frac{a}{n} -} \\{p_{ON}\left( e^{\Lambda} \right)}\end{bmatrix}\frac{n}{m}{\underset{i = 1}{\sum\limits^{m}}{\nu_{i}{e^{\Lambda - v_{i}}}^{e^{\Lambda}}}}}{{p_{ON}\left( e^{\Lambda} \right)}\left\lbrack {1 - {p_{ON}\left( e^{\Lambda} \right)}} \right\rbrack}}} & (k) \\{L^{''} = {{{L^{\prime}(\Lambda)}{e^{\Lambda}\left\lbrack {1 - \frac{\begin{matrix}{\frac{1}{m}\underset{i = 1}{\sum\limits^{m}}} \\{v_{i}{e^{- v_{i}}}^{e^{\Lambda}}}\end{matrix}}{\begin{matrix}{p_{ON}\left( e^{\Lambda} \right)} \\\left\lbrack {1 - {p_{ON}\left( e^{\Lambda} \right)}} \right\rbrack\end{matrix}} - \frac{\underset{i = 1}{\sum\limits^{m}}v_{i}^{2}}{\underset{i = 1}{\sum\limits^{m}}{v_{t}e^{v_{i}}}}} \right\rbrack}} - \frac{n{e^{2\Lambda}\left( {\frac{1}{m}{\underset{i = 1}{\sum\limits^{m}}{v_{i}{e^{- v_{i}}}^{e^{\Lambda}}}}} \right)}^{2}}{\begin{matrix}{p_{ON}\left( e^{\Lambda} \right)} \\\left\lbrack {1 - {p_{ON}\left( e^{\Lambda} \right)}} \right\rbrack\end{matrix}}}} & (l)\end{matrix}$

To maximize L(Λ), the root of L′(Λ) is found (equation (2), which isrepeated below), and it is verified that it corresponds to a maximum bychecking the sign of the second derivative (equation (m)). Aninteresting observation is that equation (2) can be obtained by usinga/n to estimate p_(ON) (λ_(T))

$\begin{matrix}{0 = {{L^{\prime}(\Lambda)} = {{\frac{a}{n} - 1 + {\frac{1}{m}\left( {\underset{i = 1}{\sum\limits^{m}}e^{{- v_{i}}\lambda_{T}}} \right)}} = {{\frac{a}{n} - {p_{ON}\left( e^{\Lambda} \right)}} = {\frac{a}{n} - {p_{ON}\left( \lambda_{T} \right)}}}}}} & (2) \\{\mspace{79mu}{\left. {L^{''}(\Lambda)} \right|_{\Lambda = \overset{\hat{}}{\Lambda}} = {\left. {L^{''}(\Lambda)} \right|_{{L^{\prime}{(\Lambda)}} = 0} = {{0 - \frac{n{e^{2\Lambda}\left( {\frac{1}{m}{\underset{i = 1}{\sum\limits^{m}}{v_{i}{e^{- v_{i}}}^{e^{\Lambda}}}}} \right)}^{2}}{\begin{matrix}{p_{ON}\left( e^{\Lambda} \right)} \\\left\lbrack {1 - {p_{ON}\left( e^{\Lambda} \right)}} \right\rbrack\end{matrix}}} < 0}}}} & (m)\end{matrix}$

Then σ_({circumflex over (Λ)}) is calculated using the expected Fisherinformation, −<L″(Λ)>. The second derivative, L″(Λ), is a linearfunction of

$\begin{matrix}{{\frac{a}{n} - {{p_{ON}\left( e^{\Lambda} \right)}\left\langle {\frac{a}{n} - {p_{ON}\left( e^{\Lambda} \right)}} \right\rangle}} = 0.} & \left( \left( {{equation}\mspace{14mu} 2} \right) \right)\end{matrix}$

can be plugged into equation (k), and the subsequent result can beplugged into equation (l) to obtain the following expression forσ_({circumflex over (Λ)}).

$\begin{matrix}{\sigma_{\overset{\hat{}}{\Lambda}} = {\sqrt{variance} = {\sqrt{\frac{1}{- \left\langle \begin{matrix}L^{''} \\(\Lambda)\end{matrix} \right\rangle}} = {\frac{1}{\begin{matrix}\left( e^{\Lambda} \right) \\\begin{pmatrix}{\frac{1}{m}\underset{i = 1}{\sum\limits^{m}}} \\{v_{i}{e^{- v_{i}}}^{e^{\Lambda}}}\end{pmatrix}\end{matrix}}\sqrt{\frac{\begin{matrix}{{p_{on}\left( e^{\Lambda} \right)} \cdot} \\\begin{bmatrix}{1 -} \\{p_{on}\left( e^{\Lambda} \right)}\end{bmatrix}\end{matrix}}{n}}}}}} & (n)\end{matrix}$

In this particular case, the standard error calculated using theobserved Fisher information, −L″(Λ), is also the same as equation (n)evaluated at Λ={circumflex over (Λ)}. This can be verified by pluggingL′({circumflex over (Λ)})=0 into L″({right arrow over (Λ)}) (equation(l)).

EXAMPLE EMBODIMENTS

Example 1 includes a system, comprising: a device configured to generatecompartments of a sample, at least one of the compartments having arespective volume that is different from a respective volume of each ofat least another one of the compartments; a detector configured todetermine a number of the compartments each having a respective numberof a target that is greater than a threshold number of the target; andelectronic circuitry configured to determine a bulk concentration of thetarget in a source of the sample in response to the determined number ofthe compartments.

Example 2 includes the system of Example 1 wherein the device includes acontainer configured to generate the compartments of the sample in abarrier phase in response to the container moving.

Example 3 includes the system of any of Examples 1-2 wherein the deviceincludes a container configured to generate the compartments of thesample in a liquid in response to a shaking of the container.

Example 4 includes the system of any of Examples 1-3 wherein the deviceincludes a container configured to generate the compartments of thesample as droplets of the sample in an oil in response to a shaking ofthe container.

Example 5 includes the system of any of Examples 1-4 wherein the deviceincludes a container configured to generate the compartments of thesample as droplets of the sample in a barrier phase in response to ashaking of the container, the droplets each have a viscosity, thebarrier phase having a viscosity that is greater than the viscosity ofthe droplets.

Example 6 includes the system of any of Examples 1-5 wherein thedetector is configured to determine the number of the compartments eachhaving a respective number of the target that is greater than thethreshold number of the target in response to a wavelength ofelectromagnetic energy at which each of the number of the compartmentsluminesces.

Example 7 includes the system of any of Examples 1-6 wherein thedetector is configured to determine the number of the compartments eachhaving a respective number of the target that is greater than thethreshold number of the target in response to a wavelength ofelectromagnetic energy that each of the number of the compartmentsabsorbs.

Example 8 includes the system of any of Examples 1-7 wherein thedetector is configured to determine the number of the compartments eachhaving a respective number of the target that is greater than thethreshold number of the target in response to a wavelength ofelectromagnetic energy that each of the number of the compartmentspasses.

Example 9 includes the system of any of Examples 1-8 wherein thedetector is configured to determine the number of the compartments eachhaving a respective number of the target that is greater than thethreshold number of the target in response to a wavelength ofelectromagnetic energy that each of the number of the compartmentsblocks.

Example 10 includes the system of any of Examples 1-9 wherein theelectronic circuitry is configured to determine a bulk concentration ofthe target in a source of the sample in response to a respectivemeasured volume of each of the number of compartments.

Example 11 includes the system of any of Examples 1-10 wherein theelectronic circuitry is configured to determine a bulk concentration ofthe target in a source of the sample in response to a sum of arespective measured volume of each of the compartments.

Example 12 includes the system of any of Examples 1-11 wherein theelectronic circuitry is configured to determine a bulk concentration ofthe target in a source of the sample in response to a number of thecompartments.

Example 13 includes the system of any of Examples 1-12 wherein theelectronic circuitry is configured to determine a bulk concentration ofthe target in a source of the sample in response to a number of othercompartments.

Example 14 includes the system of any of Examples 1-13 wherein theelectronic circuitry is configured to determine a bulk concentration ofthe target in a source of the sample in response to a respectivemeasured volume of each of other compartments.

Example 15 includes the system of any of Examples 1-14 wherein theelectronic circuitry is configured to determine a bulk concentration ofthe target in a source of the sample in response to a number of othercompartments and a respective measured volume of each of the othercompartments.

Example 16 includes the system of any of Examples 1-15 wherein theelectronic circuitry is configured to determine a bulk concentration ofthe target in a source of the sample in response to a probabilitydensity function of compartment volume.

Example 17 includes a system, comprising: a barrier-phase liquid; acontainer configured to receive the barrier-phase liquid, to receive asample including a target, and to generate compartments of the samplesuspended in the barrier-phase liquid in response to a shaking of thecontainer, at least one of the compartments having a respective volumethat is different from a respective volume of each of at least anotherone of the compartments; and a detector configured to determine a numberof the compartments each having a respective number of the target thatis greater than a threshold number of the target.

Example 18 includes the system of Example 17 wherein the barrier-phaseliquid includes an oil.

Example 19 includes the system of any of Examples 17-18 wherein thecontainer includes a clear tube.

Example 20 includes the system of any of Examples 17-19 wherein thedetector includes an electronic detector.

Example 21 includes the system of any of Examples 17-20 wherein thedetector is configured to determine a number of the compartments.

Example 22 includes the system of any of Examples 17-21, furthercomprising an apparatus configured to obtain the sample from a sourceincluding the target.

Example 23 includes the system of any of Examples 17-22, furthercomprising a computer-readable medium storing instructions that, whenexecuted by a computing circuit, cause the computing circuit todetermine a bulk concentration of the target in a source of the samplein response to the number of the compartments each having a respectivenumber of the target that is greater than a threshold number of thetarget.

Example 24 includes a method, comprising: generating compartments of asample, at least one of the compartments having a respective volume thatis different from a respective volume of each of at least another one ofthe compartments; determining a number of the compartments each having arespective number of a target that is greater than a threshold number ofthe target; and determining a bulk concentration of the target in asource of the sample in response to the number of the compartments.

Example 25 includes the method of Example 24 wherein generating thecompartments includes generating the compartments suspended in a barrierphase by shaking a container that includes the sample and the barrierphase.

Example 26 includes the method of any of Examples 24-25 whereingenerating the compartments includes generating droplets suspended in aliquid by shaking a container that includes the sample and the liquid.

Example 27 includes the method of any of Examples 24-26 whereindetermining the number of compartments each having a respective numberof the target that is greater than the threshold number of the targetincludes determining the number of compartments in response to awavelength of electromagnetic energy at which each of the number of thecompartments luminesces.

Example 28 includes the method of any of Examples 24-27 whereindetermining the number of the compartments each having a respectivenumber of the target that is greater than the threshold number of thetarget includes determining the number of compartments in response to awavelength of electromagnetic energy that each of the number of thecompartments absorbs.

Example 29 includes the method of any of Examples 24-28 whereindetermining the number of the compartments each having a respectivenumber of the target that is greater than the threshold number of thetarget includes determining the number of compartments in response to awavelength of electromagnetic energy that each of the number of thecompartments passes.

Example 30 includes the method of any of Examples 24-29 whereindetermining the number of the compartments each having a respectivenumber of the target that is greater than the threshold number of thetarget includes determining the number of compartments in response to awavelength of electromagnetic energy that each of the number of thecompartments blocks.

Example 31 includes the method of any of Examples 24-30 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to arespective measured volume of each of the number of compartments.

Example 32 includes the method of any of Examples 24-31 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to a sumof a respective measured volume of each of the compartments.

Example 33 includes the method of any of Examples 24-32 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to anumber of the compartments.

Example 34 includes the method of any of Examples 24-33 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to anumber of other compartments.

Example 35 includes the method of any of Examples 24-34 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to arespective measured volume of each of other compartments.

Example 36 includes the method of any of Examples 24-35 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to anumber of other compartments and a respective measured volume of each ofthe other compartments.

Example 37 includes the method of any of Examples 24-36 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to aprobability density function of compartment volume.

Example 38 includes a tangible non-transitory computer-readable mediumstoring instructions that, when executed by a computing circuit, causethe computing circuit: to determine a number of compartments of a sampleeach having a respective number of a target that is greater than athreshold number of the target, at least one of the compartments havinga respective volume that is different from a respective volume of eachof at least another one of the compartments; and to determine a bulkconcentration of the target in a source of the sample in response to thedetermined number of compartments.

From the foregoing it will be appreciated that, although specificembodiments have been described herein for purposes of illustration,various modifications may be made without deviating from the spirit andscope of the disclosure. Furthermore, where an alternative is disclosedfor a particular embodiment, this alternative may also apply to otherembodiments even if not specifically stated. In addition, any describedcomponent or operation may be implemented/performed in hardware,software, firmware, or a combination of any two or more of hardware,software, and firmware. For example, any of one, more, or all of theabove-described operations and functions can be performed by electroniccircuitry that is hardwire configured to perform one or more operationsor functions, that is configured to execute program instructions toperform one or more operations or functions, that is configured withfirmware, or otherwise configured, to perform one or more operations orfunctions, or that is configured with a combination of two or more ofthe aforementioned configurations. For example, one or more of thecomponents of the computer 54 of FIG. 11 can include such electroniccircuitry. Furthermore, one or more components of a described apparatusor system may have been omitted from the description for clarity oranother reason. Moreover, one or more components of a describedapparatus or system that have been included in the description may beomitted from the apparatus or system. In addition, one or more steps ofa described method may have been omitted from the description forclarity or another reason. Moreover, one or more steps of a describedmethods that have been included in the description may be omitted fromthe method.

1-23. (canceled)
 24. A method executed by a computing circuit,comprising: generating compartments of a sample, at least one of thecompartments having a respective volume that is different from arespective volume of each of at least another one of the compartments;determining a characterized number of the compartments each having arespective number of a target that is greater than a threshold number ofthe target and a characterized volume for each of the compartmentshaving the target greater than the threshold number; and determining abulk concentration of the target in a source of the sample by generatingan estimate of the bulk concentration in response to the characterizednumber of the compartments having the target greater than the thresholdnumber and the characterized volume for each of the compartments havingthe target greater than the threshold number, wherein the bulkconcentration is determined by a digital-variable-volume-approximation(DVVA) algorithm.
 25. The method of claim 24 wherein generating thecompartments includes generating the compartments suspended in a barrierphase by shaking a container that includes the sample and the barrierphase.
 26. The method of claim 24 wherein generating the compartmentsincludes generating droplets suspended in a liquid by shaking acontainer that includes the sample and the liquid.
 27. The method ofclaim 24 wherein determining the characterized number of compartmentseach having a respective number of the target that is greater than thethreshold number of the target includes determining the characterizednumber of compartments in response to a wavelength of electromagneticenergy at which each of the number of the compartments luminesces. 28.The method of claim 24 wherein determining the characterized number ofthe compartments each having a respective number of the target that isgreater than the threshold number of the target includes determining thecharacterized number of compartments in response to a wavelength ofelectromagnetic energy that each of the number of the compartmentsabsorbs.
 29. The method of claim 24 wherein determining thecharacterized number of the compartments each having a respective numberof the target that is greater than the threshold number of the targetincludes determining the characterized number of compartments inresponse to a wavelength of electromagnetic energy that each of thenumber of the compartments passes.
 30. The method of claim 24 whereindetermining the characterized number of the compartments each having arespective number of the target that is greater than the thresholdnumber of the target includes determining the characterized number ofcompartments in response to a wavelength of electromagnetic energy thateach of the number of the compartments blocks.
 31. The method of claim24 wherein determining the bulk concentration of the target in thesource of the sample includes determining the bulk concentration inresponse to a respective measured volume of each of the number ofcompartments.
 32. The method of claim 24 wherein determining the bulkconcentration of the target in the source of the sample includesdetermining the bulk concentration in response to a sum of a respectivemeasured volume of each of the compartments.
 33. The method of claim 24wherein determining the bulk concentration of the target in the sourceof the sample includes determining the bulk concentration in response toa number of the compartments.
 34. The method of claim 24 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to anumber of other compartments.
 35. The method of claim 24 whereindetermining the bulk concentration of the target in the source of thesample includes determining the bulk concentration in response to arespective measured volume of each of other compartments.
 36. The methodof claim 24 wherein determining the bulk concentration of the target inthe source of the sample includes determining the bulk concentration inresponse to a number of other compartments and a respective measuredvolume of each of the other compartments.
 37. The method of claim 24wherein determining the bulk concentration of the target in the sourceof the sample includes determining the bulk concentration in response toa probability density function of compartment volume.
 38. (canceled) 39.The method of claim 24 wherein determining the bulk concentration of thetarget in the source of the sample includes determining a bulkconcentration in response to a probability density function ofcompartment volume.